This does sound extremely odd, particularly the taking of money from your mother's funds without issuing an invoice - that is a really serious breach of the solicitors accounts rules and does lead to solicitors being struck off.
I would suggest that you start by checking exactly what sort of power of attorney he has. There are several sorts. You can grant a general power of attorney which allows people to act on your behalf. They cease immediately on someone losing mental capacity. They are often used if, for example, someone is out of the country and needs their attorney to manage a bank account or investments while they are away.
Before October 2007, you could create an Enduring Power of Attorney which could continue even after someone became mentally incapable of looking after their affairs. They were designed to give someone control of your affairs if you became unable to deal with them yourself. If your mother signed one of these pre 2007, they are still valid, and can be used, but once someone becomes mentally incapable, they have to be registered at the Court of Protection to be used.
Since October 2007, it has only been possible to create Lasting Powers of Attorney for your financial affairs, and they can be used both before and after mental incapacity and have to be registered at the Court of Protection.
The powers the solicitor has really do depend on the type of power of attorney, and whether your mother has mental capacity to deal with her own affairs.
A solicitor acting under a power of attorney may have been appointed as a solicitor of a particular firm or could have been appointed in his personal capacity - you need to ask Firm A which it was. You need to find out whether the firm generally was acting for your mother - if so, they are most definitely liable if he has done something dodgy, and will need to contact their insurers as your mother may well have a claim against them.
It is fine, and fairly usual, where a partner moves to another firm, for him to take his clients with him, particularly if he is acting as attorney. However, your mother should have been asked whether she was happy with this, and should have given her consent. If she had refused, nothing should have left the first firm.
I realise you are in a very difficult position if you suspect that something untoward has gone on, particularly if your mother does not want to make a complaint or rock the boat. As a first port of call, I think you should speak to Firm A, ask for a copy of the complaints proceedure and say you want to speak to the senior partner to discuss this. They do have to talk to you if your mother has given you authority to discuss her affairs, but they may need a letter from her confirming this (unless she has lost capacity to deal with her own affairs and you are her attorney under an Enduring Power of Attorney or Lasting Power of Attorney which has been registered with the Court of Protection - in which case you have authority to deal with her affairs generally unless there are restrictions in the Enduring or Lasting Power of Attorney).
Depending on what you find out, you may then need to go to Firm B and again ask to speak to the senior partner. I hope you manage to find some resolution to this - powers of attorney are very powerful and unfortunately can be open to abuse.
If monies have gone from your mother's funds and there is a real issue here, if either Firm A or Firm B does not deal with it - and they should - it is a huge breach of professional practice if they ignore it, your next port of call is to contact the Solicitors Regulation Authority. They regulate solicitors firms and have the power to strike off solicitors and indeed close down firms.
To be honest, if things do start looking as if there has been dishonesty by the solicitor, I would recommend that you and your mother go and get some legal advice.